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A323451 Number of ways to fill a Young diagram with positive integers summing to n such that all rows and columns are strictly increasing. 3
1, 1, 1, 3, 3, 6, 9, 12, 19, 27, 39, 54, 79, 107, 150, 209, 282, 387, 525, 707, 949, 1272, 1688, 2244, 2968, 3902, 5125, 6712, 8752, 11383, 14780, 19109, 24671, 31768, 40791, 52280, 66860, 85296, 108621, 138054, 175085, 221676, 280161, 353414, 445098, 559661 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..50

nLab, Young Diagram.

The Unapologetic Mathematician weblog, Generalized Young Tableaux.

EXAMPLE

The a(8) = 19 generalized Young tableaux:

  8   1 7   2 6   3 5   1 2 5   1 3 4

.

  1   2   3   1 2   1 5   1 3   1 4   2 3   1 2   1 2 3

  7   6   5   5     2     4     3     3     2 3   2

.

  1   1   1 2

  2   3   2

  5   4   3

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

sqfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[sqfacs[n/d], Min@@#>=d&]], {d, Select[Rest[Divisors[n]], SquareFreeQ]}]];

ptnplane[n_]:=Union[Map[primeMS, Join@@Permutations/@sqfacs[n], {2}]];

Table[Sum[Length[Select[ptnplane[Times@@Prime/@y], And@@(LessEqual@@@Transpose[PadRight[#]/.(0->Infinity)])&&And@@(UnsameQ@@@DeleteCases[Transpose[PadRight[#]], 0, {2}])&]], {y, IntegerPartitions[n]}], {n, 10}]

CROSSREFS

Cf. A000085, A000219, A003293, A114736, A117433, A138178, A299968.

Cf. A323430, A323436, A323437, A323439, A323450.

Sequence in context: A136289 A128012 A289143 * A280240 A058628 A035528

Adjacent sequences:  A323448 A323449 A323450 * A323452 A323453 A323454

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 16 2019

EXTENSIONS

a(21)-a(45) from Seiichi Manyama, Aug 19 2020

STATUS

approved

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Last modified June 20 20:36 EDT 2021. Contains 345235 sequences. (Running on oeis4.)